Law of Large Numbers (LLN) and Central Limit Theorem (CTL)

30th June 2015

Law of Large Numbers (LLN) and Central Limit Theorem (CTL)

Law of Large Numbers (LLN)

Definition

Be \(X_1, X_2, ...\) random variables with finite variances and be \(S_n = X_1+X_2+...+X_n\). The sequence \({X_n:n\geq1}\) satisfies the Law of Large Numbers if:

\(\frac{S_n}{n}-E(\frac{S_n}{n}) \xrightarrow{\mathbb{P}r}0\)

(Magalhães 2011)

http://vis.supstat.com/2013/04/law-of-large-numbers/

Uniform Distribution Example

Binomial Distribution Example

Exponential Distribution Example

Poisson Distribution Example

Cauchy Distribution Example

Central Limit Theorem (CLT)

Definition

Be \({X_n:n\geq1}\) independent random variables, identically distributed and with expected value \(\mu\) and variance \(\sigma^2\), with \(0<\sigma^2<\infty\). For \(S_n=X_1+X_2+...+X_n\),

\(\dfrac{S_n-n\mu}{{\sqrt{n\sigma^2}}} \xrightarrow{\mathbb{D}} N(0,1)\)

(Magalhães 2011)

https://www.youtube.com/watch?v=epq-dpMJIxs

Uniform Distribution Example

Binomial Distribution Example

Beta Distribution Example

Exponential Distribution Example